Multiple phase transitions in long-range first-passage percolation on square lattices
We consider a model of long-range first-passage percolation on the d-dimensional square lattice in which any two distinct vertices x, y are connected by an edge having exponentially distributed passage time with mean d(x,y)s, where s is a fixed positive parameter and d( , ) is the l1-distance. We analyze the asymptotic growth rate of the set Bt, which consists of all x∈ Zd such that x can be reached starting from 0 within time t, as t → ∞. We show that depending on the value of s there are four growth regimes:
(i) instantaneous growth for s<d,
(ii) stretched exponential growth for s ∈(d,2d),
(iii) superlinear growth for s ∈(2d,2d+1) and finally
(iv) linear growth for s>2d+1 like the nearest-neighbor first-passage percolation model corresponding to s=∞.
We will find explicit growth rates and also analyze the behavior at the boundary values of s.
The classical contact process on Zd is one of the simplest model for the propagation of an infection in a (non-moving) population. The process starts at time 0 with all sites healthy, excepted the site at the origine which is infected. Then, sites are infected with a rate proportional to their number of infected neighbours, and recover with constant rate one. We are interested in the characteristics of the growth of the infection, when the infection survives.
Durrett and Griffeath proved an asymptotic shape theorem for this model: conditionaly on the survival of the infection, the set of sites that have been infected at least once before time t asymptotically grows like tA, where A is a deterministic non-empty compact convex set of Rd.
In this talk, we explore the properties of the contact process on Zd in random environment: the proportionality constant governing the rate of infection is no longer constant, but is randomly chosen for each edge of Zd. In this more complex context, the classical proof for the shape theorem is no longer valid. We introduce a new quantity that we call the essential hitting time and that captures the regenerating structure of the surviving contact process. With this quantity, we can prove a shape theorem and also associated large deviations inequalities.
Our approach offers an alternative proof for the shape theorem in the classical case.